This happened to me as well, where my WMS providers were all working fine, but GEE was having trouble with the flatfiles when using a proxy.
The culprit in my case was that a library I used to build my proxy was automatically appending a "Content-Encoding: gzip" header to ALL content. After fixing this, your flatfiles should load just fine. Alternatively, assuming it's an option, just don't use a proxy and instead modify the Access-Control-Allow-Origin property on the GEE server, as Rachel suggests.
Steps to debug: check out the network tab in Chrome and compare the response headers you get through your proxy vs. the ones you get normally.
Hope that helps.
Justin